The Kronecker product has some properties as the wikipedia http://en.wikipedia.org/wiki/Kronecker_product.
For the sake of simplicity, we denote $\mathbf{U}_{M}^T\otimes \cdots\mathbf{U}_{m+1}^T\otimes \mathbf{U}_{m-1}^T\otimes\cdots\otimes \mathbf{U}_{1}^T$ as $\bigotimes_{k=M,k\neq m}^{1}\mathbf{U}_{k}^T$, where $\mathbf{U}_k\in \mathbb{R}^{I_k\times J_k}$.
Could anyone prove the following fomulations? $$ \bigotimes_{k=M,k\neq m}^{1}\mathbf{U}_{k}^T=\bigotimes_{k=1,k\neq m}^{M}\mathbf{U}_{k}^T $$ $$ \left(\bigotimes_{k=1,k\neq m}^{M}\mathbf{U}_{k}^T\right)^T=\bigotimes_{k=1,k\neq m}^{M}\mathbf{U}_{k} $$
Proof:
The first formula is false. $$ \mathbf{U}_1=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$ $$ \mathbf{U}_3=\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} $$ $$ \mathbf{U}_1^T\otimes \mathbf{U}_3^T \neq \mathbf{U}_3^T\otimes \mathbf{U}_1^T $$ The second formula is proven as follows: $$ \left(\bigotimes_{k=1,k\neq m}^{M}\mathbf{U}_k^T\right)^T=\left(\bigotimes_{k=1,k\neq m}^{M-1}\mathbf{U}_k^T\right)^T\otimes \mathbf{U}_M=\ldots=\otimes_{k=1,k\neq m}^{M}\mathbf{U}_k $$